The Temperature Dependence of Equilibrium Constant

For isothermal-isobaric reactions, the following relationships can be written  

The value of the derivative ((0Tp /07 )p) can be determined from the van t Hoff isotherm equation 

Substituting this value to the Gibbs-Helmholtz equation, the following relationship is obtained 

A reaction direction can be read from the reaction isobar equation. Under the implicit assumption during exothermic reactions, the thermal effect has a negative sign 

If the equilibrium constant K increases with increasing temperature, the concentration of substrate increases too, and reduces the concentration of products. In this case the increase in temperature of the exothermic reaction causes a reduction in reaction performance. Exothermic reactions proceed favorably in terms of the total amount of substrate conversion into products at low temperatures. 

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The temperature dependency of equilibrium constants and of Henry s constants are compiled in tables A II.I and A II.II. 

Also, AH values are required to calculate the temperature dependence of equilibrium constants. For aU these reasons, it is desirable to have tables of AH values available, so that the enthalpies of various transformations can be calculated readily. In many of these calculations, we make use of Hess s law, which is now firmly established on the basis of the first law of thermodynamics. We can then calculate AH for reactions for which the heat effect is difficult to measure but that can be expressed as sums of reactions with known values of AH. 

Table 3.5 gives the average change in Kin per 10°C increase/decrease in temperature for various A12H, values. A much more comprehensive table which is extremely useful for assessing the temperature dependence of equilibrium constants as well as of reaction rate constants is Table D1 in Appendix D. 

So there is an underlying basis related to the standard enthalpy of reaction (or heat capacities, Eq. 2.26) for the equation form used in this work to characterize the temperature dependence of equilibrium constants and Pitzer parameters (cf. Eqs. 2.70 and 2.73). 

The equation AG° - AIT - TA5° tells us that how ACT1 varies with temperature depends mainly on the entropy change for the reaction (AS°). We need these terms to explain the temperature dependence of equilibrium constants and to explain why some reactions may absorb heat (endothermic) while others give out heat (exothermic). 

In contrast to the formally analogous van t Hoff equation [10] for the temperature dependence of equilibrium constants, the Arrhenius equation 1.3 is empirical and not exact The pre-exponential factor A is not entirely independent of temperature. Slight deviations from straight-line behavior must therefore be expected. In terms of collision theory, the exponential factor stems from Boltzmann s law and reflects the fact that a collision will only be successful if the energy of the molecules exceeds a critical value. In addition, however, the frequency of collisions, reflected by the pre-exponential factor A, increases in proportion to the square root of temperature (at least in gases). This relatively small contribution to the temperature dependence is not correctly accounted for in eqns 2.2 and 2.3. [For more detail, see general references at end of chapter.]... 

The enthalpies of reaction, Aj//°(IX.15) and A //° (1X.16), selected by the present review are based on the data of [1959Z1E] that have been obtained using calorimetry these data are much more accurate than those from the temperature dependence of equilibrium constants in [1972PAT/RAM]. The experimental data refer to a 2.2 m HCIO4 ionic medium and this review has assumed that they are a good approximation of the values at zero ionic strength. The selected entropy of reaction at... 

Equilibrium Measurements. Measurements of the temperature dependence of equilibrium constants of reactions involving transition metal-alkyl bond disruption yield values of aH from which BDE s can be deduced through appropriate thermodynamic cycles. The first example of this application involved the determination of the Co-C BDE of Py(DH)2Co-CH(CH )Ph from measurements of the equilibrium constant of reaction 17, according to Equations 17-19 (28,29). 

The phenomenon of compensation is not unique to heterogeneous catalysis it is also seen in homogeneous catalysts, in organic reactions where the solvent is varied and in numerous physical processes such as solid-state diffusion, semiconduction (where it is known as the Meyer-Neldel Rule), and thermionic emission (governed by Richardson s equation ). Indeed it appears that kinetic parameters of any activated process, physical or chemical, are quite liable to exhibit compensation it even applies to the mortality rates of bacteria, as these also obey the Arrhenius equation. It connects with parallel effects in thermodynamics, where entropy and enthalpy terms describing the temperature dependence of equilibrium constants also show compensation. This brings us the area of linear free-energy relationships (LFER), discussion of which is fully covered in the literature, but which need not detain us now. 

Enthalpies of reaction can be measured by calorimetric techniques. Alternatively, the temperature dependence of equilibrium constants can be used to determine A/f° and A5° for these ligand substitution reactions by plotting In K versus /T. 

SECTIONS 19.6 AND 19.7 The values of AH and AS generally do not vary much with temperature. Therefore, the dependence of AG with temperature is governed mainly by the value of T in the expression AG = AH — TAS. The entropy term —TAS has the greater effect on the temperature dependence of AG and, hence, on the spontaneity of the process. For example, a process for which AH > 0 and As > 0, such as the melting of ice, can be nonspontaneous (AG > 0) at low temperatures and spontaneous (AG < 0) at higher temperatures. Under nonstandard conditions AG is related to AG° and the value of the reaction quotient, Q AG = AG" + RT In Q. At equilibrium (AG = 0, Q = K), AG = —RT InkT. Thus, the standard free-energy change is directly related to the equilibrium constant for the reaction. This relationship expresses the temperature dependence of equilibrium constants. 

The van t Hoff equation for the temperature dependence of equilibrium constants (Equation 10.16) is very similar in form to the Clausius-Clapeyron equation (Equation 9.4) for the temperature dependence of vapor pressure. Give an explanation for this similarity. Hint Write the process of vaporization of a liquid as an equilibrium reaction. What is the equilibrium constant )... 

Under nonstandard conditicxis, AG is related to AG and the value of the reaction quotient, Q AG = AG — RT In Q. At equilibrium (AG = 0,Q = K ), AG° = —RTlnKfy. Thus, the standard free-energy change is directly related to the equilibrium constant for the reaction. This rdationship can be used to explain the temperature dependence of equilibrium constants. 

Validity of equation (3.58) is limited to the interval in which the empirical relationship for molar heat are valid, the precision of the correlation of molar heat values being decisive for the precision of calculated equilibrium constant data. For practical purposes this precision is usually satisfactory. For illustration. Fig. 2 shows the course of the temperature dependence of equilibrium constants for some reactions which are of significance in industry. 

Equation 7.1 also gives us insight into the temperature dependence of equilibrium constants. First, we suppose that both the forward and reverse reactions show Arrhenius behavior (Section 6.6). As we see from Fig. 7.1, for an exothermic reaction the activation energy of the forward reaction is smaller than that of the reverse reaction. Therefore, the forward rate constant increases less sharply with... 

The two most commonly employed techniques for obtaining complexation enthalpies are based on the temperature dependence of equilibrium constants or calorimetric procedures. In the latter, the heat evolved when the acid and base are mixed in the reaction cell of a calorimeter is measured. The molar enthalpy of complexation, AH°, is related to the measured heat output, Q, corrected for the heats of dilution, the equilibrium concentration of the complex, [AB], and the volume of the solution in litres, V, by the relation... 

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